Domain Markov Property Research Articles

A characterisation of the continuum Gaussian free field in arbitrary dimensions

We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Topological characterisations of Loewner traces

The (chordal) Loewner differential equation encodes certain curves in the half-plane (aka traces) by continuous real-valued driving functions. Not all curves are traces; the latter can be defined via a geometric condition called the local growth property. In this paper we give two other equivalent conditions that characterise traces: 1. A continuous curve is a trace if and only if mapping out any initial segment preserves its continuity (which can be seen as an analogue of the domain Markov property of SLE). 2. The (not necessarily simple) traces are exactly the uniform limits of simple traces. Moreover, using methods by Lind, Marshall, Rohde (2010), we infer that uniform convergence of traces imply uniform convergence of their driving functions.

(1+𝜀) moments suffice to characterise the GFF

We show that there is “no stable free field of index α∈(1,2)”, in the following sense. It was proved in [4] that subject to a fourth moment assumption, any random generalised function on a domain D of the plane, satisfying conformal invariance and a natural domain Markov property, must be a constant multiple of the Gaussian free field. In this article we show that the existence of (1+𝜀) moments is sufficient for the same conclusion. A key idea is a new way of exploring the field, where (instead of looking at the more standard circle averages) we start from the boundary and discover averages of the field with respect to a certain “hitting density” of Itô excursions.

UST branches, martingales, and multiple SLE(2)

We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple $\mathrm {SLE}(2)$, i.e., an $\mathrm {SLE}(2)$ process weighted by a suitable partition function. By recent results, this also characterizes the “global” scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with $N$ branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence proof for a boundary-visiting UST branch and a boundary-visiting $\mathrm {SLE}(2)$.

A characterisation of the Gaussian free field

We prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.

Polychromatic Arm Exponents for the Critical Planar FK-Ising Model

Schramm Loewner Evolution (SLE) is a one-parameter family of random planar curves introduced by Oded Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain "domain Markov property". In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to SLE$_{16/3}$. We derive the arm exponents of SLE$_{\kappa}$ for $\kappa\in (4,8)$. Combining with the convergence of the interface, we derive the arm exponents of the critical FK-Ising model. We obtain six different patterns of boundary arm exponents and three different patterns of interior arm exponents of the critical planar FK-Ising model on the square lattice.

Classification of half-planar maps

We characterize all translation invariant half-planar maps satisfying a certain natural domain Markov property. For $p$-angulations with $p\ge3$ where all faces are simple, we show that these form a one-parameter family of measures $\mathbb{H}^{(p)}_{\alpha}$. For triangulations, we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well-known half-plane uniform infinite planar maps. The subcritical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.

On Site Percolation in Random Quadrangulations of the Half-Plane

We study site percolation on uniform quadrangulations of the upper half plane. The main contribution is a method for applying Angel’s peeling process, in particular for analyzing an evolving boundary condition during the peeling. Our method lets us obtain rigorous and explicit upper and lower bounds on the percolation threshold \(p_\mathrm {c}\), and thus show in particular that \(0.5511\le p_\mathrm {c}\le 0.5581\). The method can be extended to site percolation on other half-planar maps with the domain Markov property.

Geometry and percolation on half planar triangulations

We analyze the geometry of domain Markov half planar triangulations. In [5] it is shown thatthere exists a one-parameter family ofmeasures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition inview of their geometry atα = 2/3. For α < 2/3, the maps form atree-like stricture with infinitely many small cut-sets.For α > 2/3,we obtain maps of hyperbolic nature with exponential growth andanchoredexpansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.

Overruled harmonic explorers in the plane and stochastic Löwner evolution

The stochastic Löwner evolutions are a family of continuous stochastic curves in the real parameter , but only few and isolated examples of two-dimensional discrete growth processes areknown to converge to a . Although one-parameter families of discrete stochastic processes are provided by spin models such as theO(n) modelor the q-state Potts model, building an exploration process on the basis of these is onerous andcomputationally expensive, because it requires solving for the entire domain at each step.The basic idea of the present work is the search for a one-parameter family ofcomputationally cheap exploration processes in one-to-one correspondence with . We introduce a class of exploration processes in the plane that extends the harmonicexplorer. These processes, dubbed overruled harmonic explorers, enjoy the domain Markovproperty by construction and are supposed to converge to in the scaling limit. We show by means of numerical simulations that crossingprobabilities in rectangular domains are indeed conformally invariant, and conjecture alinear relation between and the parameter p labelling the overruled harmonic explorer.

The covariant measure of SLE on the boundary

We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in question. Assuming that boundary measures transform in a ``d-dimensional'' way (where d is the Hausdorff dimension of gamma intersected with R), we show that the measure we construct is (up to multiplicative constant) the unique measure-valued function of the SLE path that satisfies the Domain Markov property.

Possible description of domain walls in two-dimensional spin glasses by stochastic Loewner evolutions

Domain walls for spin glasses are believed to be scale invariant; a stronger symmetry, conformal invariance, has the potential to hold. The statistics of zero-temperature Ising spin glass domain walls in two dimensions are used to test the hypothesis that these domain walls are described by a Schramm-Loewner evolution ${\mathrm{SLE}}_{\ensuremath{\kappa}}$. Multiple tests are consistent with ${\mathrm{SLE}}_{\ensuremath{\kappa}}$, where $\ensuremath{\kappa}=2.32\ifmmode\pm\else\textpm\fi{}0.08$. Both conformal invariance and the domain Markov property are tested. The latter does not hold in small systems, but detailed numerical evidence suggests that it holds in the continuum limit.